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The method of detection and localization of configuration defects in geodetic networks by means of Tikhonov regularization

Roman Kadaj
Roman Kadaj
   | 17 déc. 2021
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Article Category: Original Article
Publié en ligne: 17 déc. 2021
Pages: 19 - 25
Reçu: 22 juil. 2021
Accepté: 08 nov. 2021
DOI: https://doi.org/10.2478/rgg-2021-0004
Mots clés
© 2021 Roman Kadaj et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Defects in geodetic network: outliers and geometric (datum or configuration) defects.
Defects in geodetic network: outliers and geometric (datum or configuration) defects.

Figure 2

Structure of the R matrix (0 – elements assumed to be zero, X – other elements of the matrix), r – rank of the normal matrix B, u = n–r = defect of the normal matrix.
Structure of the R matrix (0 – elements assumed to be zero, X – other elements of the matrix), r – rank of the normal matrix B, u = n–r = defect of the normal matrix.

Figure 3

A symbolic sketch of an extensive geodetic network in a district in Poland. In the Polish classification, it is a third-class network related to the network of classes I and II. The sketch shows a place in the network where configuration defects have been artificially introduced.
A symbolic sketch of an extensive geodetic network in a district in Poland. In the Polish classification, it is a third-class network related to the network of classes I and II. The sketch shows a place in the network where configuration defects have been artificially introduced.

Figure 4

Localized configuration defects (numerical results in Table 1).
Localized configuration defects (numerical results in Table 1).

Fragment of the result set (coordinates and accuracy parameters) of the network adjustment containing configuration defects using Tikhonov regularization with the parameter = 1/10,000 (assuming a priori standard deviations of the coordinates μx = μy = 100 m)

Point No x [m] y [m] s [m] a [m] b [m] θ[m]
11351051 5734128.7313 7593963.7917 0.0229 0.0165 0.0159 114.89
11351052 5733836.1122 7593815.7940 0.0337 0.0265 0.0207 125.33
11351031 5734659.9622 7593795.2190 0.0338 0.0289 0.0175 47.35
11351030 5734794.1818 7593605.8420 0.0555 0.0500 0.0241 43.80
11351029 5734961.8797 7593605.2604 0.0691 0.0633 0.0277 57.73
11351028 5735150.9054 7593600.7769 0.0883 0.0820 0.0329 69.19
11351027 5735344.5530 7593600.1071 99.9993 99.9993 0.0506 99.78
11351026 5735593.1898 7593599.2057 90.0123 90.0123 0.0558 99.77
11351025 5735845.2708 7593593.3791 43.5952 43.5952 0.0460 101.09
11351024 5736163.6210 7593598.8395 0.0848 0.0792 0.0302 74.12
11351023 5736508.7176 7593598.4821 0.0569 0.0512 0.0249 55.53
11351022 5736792.5664 7593337.1697 0.0276 0.0214 0.0174 61.38
11351112 5737672.6161 7592856.1053 0.0273 0.0207 0.0178 56.84
11351102 5737876.8970 7592817.7828 0.0304 0.0232 0.0196 49.43
11351101 5738021.9520 7592509.6190 0.0357 0.0276 0.0226 49.04

Fragment of the result set (corresponding to the data in Table 1) without defects and without Tikhonov regularization

Point No x [m] y [m] s [m] a [m] b [m] θ[m]
11351051 5734128.7307 7593963.7936 0.0209 0.0153 0.0143 20.80
11351052 5733836.1113 7593815.7965 0.0310 0.0237 0.0199 126.34
11351031 5734659.9605 7593795.2131 0.0263 0.0208 0.0162 53.54
11351030 5734794.1760 7593605.8299 0.0389 0.0322 0.0218 48.54
11351029 5734961.8766 7593605.2400 0.0445 0.0379 0.0234 58.27
11351028 5735150.9048 7593600.7438 0.0500 0.0431 0.0254 66.20
11351027 5735344.5553 7593600.1044 0.0538 0.0467 0.0266 71.41
11351026 5735593.1955 7593599.3810 0.0558 0.0488 0.0270 74.54
11351025 5735845.2670 7593593.0363 0.0545 0.0477 0.0263 74.50
11351024 5736163.6139 7593598.8498 0.0487 0.0421 0.0245 69.68
11351023 5736508.7137 7593598.4877 0.0392 0.0324 0.0221 57.19
11351022 5736792.5638 7593337.1718 0.0230 0.0166 0.0160 87.27
11351112 5737672.6161 7592856.1053 0.0262 0.0199 0.0171 56.84
11351102 5737876.8970 7592817.7828 0.0292 0.0223 0.0189 49.43
11351101 5738021.9520 7592509.6190 0.0343 0.0265 0.0217 49.04
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